The Riesz transform and quantitative rectifiability for general Radon measures

In this paper we show that if $\mu$ is a Borel measure in ${\mathbb{R}}^{n+1}$ with growth of order $n$, so that the $n$-dimensional Riesz transform $R_{\mu }$ is bounded in $L^2(\mu )$, and $B\subset {\mathbb{R}}^{n+1}$ is a ball with $\mu (B)\approx r(B{)}^n$ such that: (a) there is some $n$-plane $L$ passing through the center of $B$ such that for some $\delta >0$ small enough, it holds ${\int }_B\frac{dist(x,L)}{r(B)}d\mu (x)\le \delta \mu (B),$ (b) for some constant $ϵ>0$ small enough, $\int_B |R_\mu1(x) - m_{\mu,B}(R_\mu1)|^2,d\mu(x) \leq \epsilon ,\mu(B)$,where$m_{\mu,B}(R_\mu1)$standsforthemeanof$R_\mu1$on$B$withrespectto$\mu$;thenthereexistsauniformly$n$-rectifiablesubset$\Gamma$,with$\mu(\Gamma\cap B)\gtrsim \mu(B)$,andsothat$\mu|_\Gamma$ is absolutely continuous with respect to $H^n{\mathrm{\mid }}_{\mathrm{\Gamma }}$. This result is an essential tool to solve an old question on a two phase problem for harmonic measure in a subsequent paper by Azzam, Mourgoglou and Tolsa.